A characterization of algebras with polynomial growth of the codimensions

Let be an associative algebras over a field of characteristic zero. We prove that the codimensions of are polynomially bounded if and only if any finite dimensional algebra with has an explicit decomposition into suitable subalgebras; we also give a decomposition of the -th cocharacter of into suitable -characters.

We give similar characterizations of finite dimensional algebras with involution whose -codimension sequence is polynomially bounded. In this case we exploit the representation theory of the hyperoctahedral group.

     

Polynomial Identities of Finite Dimensional Simple Algebras

Let F be an algebraically closed field and let A and B be arbitrary finite dimensional simple algebras over F. We prove that A and B are isomorphic if and only if they satisfy the same identities.     

On Codimension Growth of Finitely Generated Associative Algebras

LetAbe a PI-algebra over a fieldF. We study the asymptotic behavior of the sequence of codimensionsc_n(A) ofA. We show that ifAis finitely generated overFthenInv(A)=lim_n→∞ always exists and is an integer. We also obtain the following characterization of simple algebras:Ais finite dimensional central simple overFif and only ifInv(A)=dim=A.     

The exponential growth of codimensions for Capelli identities

By applying the theorem that every positive integer is a sum of four squares, we calculate the exponential growth of the codimensions for the relatively free algebra satisfying Capelli identities. Work partially supported by RFFI grants 96-01-00146 and 98-01-01020. Work partially supported by ISF grant 6629/1. Work partially supported by RFFI grants 96-01-00146 and 96-15-96050.     

Central polynomials and growth functions

The growth of central polynomials for the algebra of n × n matrices in characterstic zero was studied by Regev in [13]. Here we study the growth of central polynomials for any finite-dimensional algebra over a field of characteristic zero. For such an algebra A we prove the existence of two limits called the central exponent and the proper central exponent of A. They give a measure of the exponential growth of the central polynomials and the proper central polynomials of A. We study the range of such limits and we compare them with the PI-exponent of the algebra.     

-identities on associative algebras

Let be an algebra over a field and a finite group of automorphisms and anti-automorphisms of . We prove that if satisfies an essential -polynomial identity of degree , then the -codimensions of are exponentially bounded and satisfies a polynomial identity whose degree is bounded by an explicit function of . As a consequence we show that if is an algebra with involution satisfying a -polynomial identity of degree , then the -codimensions of are exponentially bounded; this gives a new proof of a theorem of Amitsur stating that in this case must satisfy a polynomial identity and we can now give an upper bound on the degree of this identity.

     

Group Actions and Asymptotic Behavior of Graded Polynomial Identities

Let F be an algebraically closed field of characteristic 0,and let A be a G-graded algebra over F for some finite abeliangroup G. Through G being regarded as a group of automorphismsof A, the duality between graded identities and G-identitiesof A is exploited. In this framework, the space of multilinearG-polynomials is introduced, and the asymptotic behavior ofthe sequence of G-codimensions of A is studied. Two characterizations are given of the ideal of G-graded identitiesof such algebra in the case in which the sequence of G-codimensionsis polynomially bounded. While the first gives a list of G-identitiessatisfied by A, the second is expressed in the language of therepresentation theory of the wreath product G S_n, where S_nis the symmetric group of degree n. As a consequence, it is proved that the sequence of G-codimensionsof an algebra satisfying a polynomial identity either is polynomiallybounded or grows exponentially.     

An example of a variety of linear algebras with fractional-polynomial growth

The asymptotic behavior of codimensions of identities of nonassociative algebras is studied in the paper. It is shown that in contrast with the associative case a polynomial growth of codimensions can result in a fractional exponent of a polynomial degree.